As discussed in the note on grazing goats, a differentiable function
y(x) can be inverted at any given point x0, y0 to give
a power series for the function x(y). Writing this power series as

it’s clear that the coefficients cn are related
to the derivatives of x with respect to y by

Hence in order to determine the coefficients of the power
series of this inverted function we need only determine the derivatives of x
with respect to y. Our task is to determine these derivatives from the known
derivatives of y with respect to x. We have y0 = y(x0).
Letting positive integer subscripts on either of these variables denote
derivatives with respect to the other variable, we saw in the previous note that
the derivatives are related by

Using this formula we can determine the expressions

and so on. For a simple example, suppose we wish to determine
the power series for the inverse of the exponential function y = ex.
In this case yn = ex for all n, and substitution into
the preceding expressions gives

and so on. We choose to expand the series about the point
x0 = 0, y0 = 1, so we evaluate these derivatives at x0
= 0, and we find that

Substituting these expressions into the equations for the
coefficients, we get the series

which of course is the power series for the natural log,
the inverse of the exponential function.

Notice that the numerators of the expression for xn
contains a term for each partition of m = 2(n-1)
into exactly (n-1) parts. Also, the overall
signs alternate for successive n, so we can write the general coefficient of
the inverse function in the form

where the summation is evaluated over all sets of
non-negative integers a1, a2, …, an such
that

The “C” coefficients in the preceding expression are
related to the multinomial and binomial coefficients. Specifically, we have

To illustrate, we will use these formulas to determine the
general expression for the 6th order coefficient in the power series of the
inverse function. The seven partitions of m = 2(n-1) = 10 into exactly (n-1)
= 5 parts are listed below, along with the corresponding C coefficients.

To illustrate how the coefficients are computed, the
coefficient corresponding to the second of these partitions is

Notice that the sum of the coefficients for n = 6 is (n-1)! = 120, as required. From these results
we can now write the sixth order coefficient in the power series expansion of
the general inverse function:

Naturally these results are consistent with the “reversion
of series” formulas, which apply in cases when the original function is given
as a power series and we wish to find the power series of the inverse
function. We can always shift the variables so that we are expanding about
the point x0 = y0 = 0, and as always we require that
the derivative of y(x) be non-zero, so we are given a series of the form

and we wish to determine the coefficients of the reverted
series

Inserting the values of the derivatives y1 = h1,
y2 = 2h2, y3 = 6h3, etc., into
the previous formulas, we get

and so on. In these terms the coefficients in the
numerators sum to unity. In the note on grazing goats we considered the
function y = sin(x) – x cos(x) at the point x0 = p/2, y0 = 1, so we could just as
well have proceeded by re-writing the function in terms of the re-scaled
parameters X = x – p/2 and Y = y - 1, which gives

Then application of the preceding formula for series
reversion gives

in agreement with the result found by evaluating the
derivatives.

It’s interesting that the condition that must be met by a
function y(x) in order for the third derivative of x(y) to vanish is y1y3
– (1/3)y22 = 0 which, if we set s0 = y1
and differentiate again, gives

This is of the same general form as the ubiquitous
“separation equation” in physics. (See the note
on inertial and gravitational separations.) This is also reminiscent of the
question about whether an accelerating
charge radiates, a question that hinges on whether radiation is
associated with the second or the third derivative of position with respect
to time.